Integrand size = 17, antiderivative size = 98 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {b^2 n^2}{4 \left (1+b^2 n^2\right ) x^2}-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right ) x^2}-\frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right ) x^2} \]
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Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4575, 30} \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (b^2 n^2+1\right )}-\frac {b n \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (b^2 n^2+1\right )}-\frac {b^2 n^2}{4 x^2 \left (b^2 n^2+1\right )} \]
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Rule 30
Rule 4575
Rubi steps \begin{align*} \text {integral}& = -\frac {b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right ) x^2}-\frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right ) x^2}+\frac {\left (b^2 n^2\right ) \int \frac {1}{x^3} \, dx}{2 \left (1+b^2 n^2\right )} \\ & = -\frac {b^2 n^2}{4 \left (1+b^2 n^2\right ) x^2}-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right ) x^2}-\frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right ) x^2} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.59 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {1+b^2 n^2-\cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}{4 \left (1+b^2 n^2\right ) x^2} \]
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Time = 2.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(\frac {-b^{2} n^{2}-b n \sin \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+\cos \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )-1}{4 x^{2} \left (b^{2} n^{2}+1\right )}\) | \(59\) |
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none
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {b^{2} n^{2} + 2 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2}{4 \, {\left (b^{2} n^{2} + 1\right )} x^{2}} \]
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Result contains complex when optimal does not.
Time = 2.44 (sec) , antiderivative size = 468, normalized size of antiderivative = 4.78 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\begin {cases} - \frac {3 i \sin {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )} \cos {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 x^{2}} - \frac {\cos ^{2}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{2 x^{2}} + \frac {\log {\left (c x^{n} \right )} \sin ^{2}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 n x^{2}} - \frac {i \log {\left (c x^{n} \right )} \sin {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )} \cos {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{2 n x^{2}} - \frac {\log {\left (c x^{n} \right )} \cos ^{2}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 n x^{2}} & \text {for}\: b = - \frac {i}{n} \\- \frac {\sin ^{2}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{2 x^{2}} - \frac {i \sin {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )} \cos {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 x^{2}} + \frac {\log {\left (c x^{n} \right )} \sin ^{2}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 n x^{2}} + \frac {i \log {\left (c x^{n} \right )} \sin {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )} \cos {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{2 n x^{2}} - \frac {\log {\left (c x^{n} \right )} \cos ^{2}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{4 n x^{2}} & \text {for}\: b = \frac {i}{n} \\- \frac {b^{2} n^{2} \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} x^{2} + 4 x^{2}} - \frac {b^{2} n^{2} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} x^{2} + 4 x^{2}} - \frac {2 b n \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} x^{2} + 4 x^{2}} - \frac {2 \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} n^{2} x^{2} + 4 x^{2}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (92) = 184\).
Time = 0.23 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.86 \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {2 \, {\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + 2 \, \cos \left (2 \, b \log \left (c\right )\right )^{2} + {\left ({\left (b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - b \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right )\right )} n - \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) - \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \cos \left (2 \, b \log \left (c\right )\right )\right )} \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) + 2 \, \sin \left (2 \, b \log \left (c\right )\right )^{2} + {\left ({\left (b \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) + b \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \cos \left (2 \, b \log \left (c\right )\right )\right )} n + \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right )\right )} \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{8 \, {\left ({\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + \cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} x^{2}} \]
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\[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{2}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3} \,d x \]
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